Skew-Hadamard matrices of orders 188 and 388 exist

TL;DR

This paper presents the first construction of skew-Hadamard matrices of orders 188 and 388 using difference families on cyclic groups and the Goethals-Seidel array, expanding the known classes of such matrices.

Contribution

First-time construction of skew-Hadamard matrices of orders 188 and 388 via difference families on cyclic groups and the Goethals-Seidel array.

Findings

Constructed difference families on cyclic groups of orders 47 and 97.

Successfully constructed skew-Hadamard matrices of orders 188 and 388.

Matrices are constructed for the first time in this work.

Abstract

We construct several difference families on cyclic groups of orders 47 and 97, and use them to construct skew-Hadamard matrices of orders 188 and 388. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array.